For ease of reference, the following documents are referred to by the following reference numbers in this document.    [1] P. Scheunders and J. Driesen, “Least-squares interband denoising of color and multispectral images,” Int'l Conf. on Image Processing, pp. 985-988, October 2004.    [2] Aleksandra Pizurica, Wilfried Philips and Paul Scheundersy, “Wavelet domain denoising of single-band and multiband images adapted to the probability of the presence of features of interest,” SPIE Conference Wavelets XI, San Diego, Calif., USA, 31 Jul.-4 Aug. 2005.    [3] Aleksandra Pizurica and Wilfried Philips, “Estimating the probability of the presence of a signal of interest in multiresolution single- and multiband image denoising,” IEEE Trans. Image Processing (in press), http://telin.ugent.be/˜sanja/Papers/TransIP2005_ProbShrink.pdf    [4] P. Scheunders, “Wavelet thresholding of multivalued images,” IEEE Trans. Image Processing, vol. 13(4), pp. 475-483, April 2004.    [5] Hyeokho Choi and R. G. Baraniuk, “Multiple wavelet basis image denoising using Besov ball projections,” IEEE Signal Processing Letters, vol. 11, issue 9, pp. 717-720, September 2004.    [6] D. L. Donoho and I. M. Johnstone, “Threshold selection for wavelet shrinkage of noisy data,” Proc. IEEE Inte'l Conf Engineering in Medicine and Biology Society, Engineering Advances: New Opportunities for Biomedical Engineers, vol. 1, pp. A24—A25, November 1994.    [7] K. S. Schmidt and A. K. Skidmore, “Smoothing vegetation Spectra with Wavelets,” Int. J. Remote Sensing, vol. 25, No. 6, pp. 1167-1184, March, 2004.    [8] M. Lang, H. Guo, J. E. Odegard, C. S. Burrus and R. O. Wells, “Non-linear processing of a shift-invariant DWT for noise reduction,” SPIE, Mathematical Imaging Wavelet Applications for Dual Use, on SPIE Symp. On OE/Aerospace Sensing and Dual Use Photonics, Orlando, Fla., 17-21 Apr. 1995.    [9] M. Lang, H. Guo, J. E. Odegard, C. S. Burrus and R. O. Wells Jr., “Noise reduction using an undecimated discrete wavelet transform,” IEEE Signal Processing Letters, vol. 3, issue 1, pp. 10-12, January 1996.    [10] T. D. Bui and G. Y. Chen, “Translation-invariant denoising using Multiwavelets,” IEEE Trans. Signal Processing, vol. 64, no. 12, pp. 3414-3420, 1998.    [11] Aglika Gyaourova, C. Kamath, and I. K. Fodor, “Undecimated wavelet transforms for image de-noising,” Lawrence Livermore National Laboratory, Livermore, Calif., Technical report, UCRL-ID-150931, Nov. 19, 2002.    [12] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelets shrinkage,” Biometrika, vol. 81, pp. 425-455, 1994.    [13] D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelets shrinkage,” J. American Statistics Association, 90(432), pp. 1200-1224, 1995.    [14] S Grace Chang, Bin Yu and Martin Vetterli, “Adaptive wavelet thresholding for image denoising and compression,” IEEE Transactions on Image Processing, vol. 9, no. 9, pp. 1532-1546, September 2000.    [15] A. G. Bruce and H. Y. Gao, “Understanding waveshrink: variance and bias estimation,” Biometrika, vol. 83, pp. 727-745, 1996.    [16] Wallace M. Poter and Harry T. Enmark, “A system overview of the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS),” SPIE, vol. 834 Image Spectroscopy II, pp. 22-30, 1987.    [17] MacDonald Dettwiller, “System studies of a small satellite hyperspectral mission, data acceptability”, Contract Technical Report to Canadian Space Agency, St-Hubert, Canada, HY-TN-51-4972, issue 2/1, Mar. 5, 2004.    [18] Shen-En Qian, Martin Bergeron, Ian Cunningham, Luc Gagnon and Allan Hollinger, “Near lossless data compression onboard a hyperspectral satellite,” IEEE Trans. Aerospace and Electronic Systems, vol. 42, no. 3, pp. 851-866, July 2006.    [19] R. Bukingham, K. Staenz and A. B. Hollinger, “Review of Canadian Airborne and Space Activities in Hyperspectral Remote Sensing,” Canadian Aeronautics and Space Journal, vol. 48, no. 1, pp. 115-121, 2002.    [20] A. Basuhail, S. P. Kozaitis, “Wavelet-based noise reduction in multispectral imagery,” SPIE Conf. Algorithms for Multispectral and Hyperspectral Imagery IV, Orlando, vol. 3372, pp. 234-240, 1998.    [21] C. Sidney Burrus, Ramesh A Gopinath and Haitao Guo, “Introduction to Wavelets and Wavelet Transforms, A primer,” Prentice Hall, 1998, pp. 88-97.    [22] Hisham Othman and Shen-En Qian, “Noise Reduction of Hyperspectral Imagery Using Hybrid Spatial-Spectral Derivative-Domain Wavelet Shrinkage,” IEEE Trans. on Geoscience and Remote Sensing, vol. 44, no. 1, pp. 397-408, February, 2006.
Satellite imagery has been used in the past for purposes as disparate as military surveillance and vegetation mapping. However, regardless of the purpose behind satellite imagery, higher quality images have always been desirable.
The reliability of the information delivered by hyperspectral remote sensing sensors (or imagers) highly depends on the quality of the captured data. Despite the advance in hyperspectral sensors, captured data carry enough noise to affect the information extraction and scene interpretation. This noise includes a signal dependent component, called photon noise, and other signal independent components, e.g. thermal noise.
A key parameter in the design of a hyperspectral imager is its Signal-to-Noise Ratio (SNR), which determines the capabilities and the cost of the imager. A sufficiently high SNR can be achieved first-hand by adopting some excessive measures in the instrument design, e.g. increasing the size of the optical system, increasing the integration time, increasing the detector area, etc. Normally, these are prohibitively expensive solutions, especially in the case of spaceborne instruments. Alternatively, modern numerical processing based Noise Reduction (NR) methods provide a cost-effective solution that is becoming more and more affordable (in terms of speed and expense) due to the availability of the advanced computing devices.
Smoothing filters and Minimum Noise Fraction (MNF) are the most popular among the legacy methods of hyperspectral/multispectral imagery NR. While smoothing filters have a negative impact on the sharp signal features, the MNF is relatively demanding in terms of computational expenses.
Several methods have been introduced recently which benefit from compactness of the wavelet transform. Examples of the recent wavelet transform-based NR methods include the Linear Minimal Mean Squared Error (LMMSE) method in [1], featuring a global and two local estimators. Although the local estimators outperform the global estimator in the color images, they suffer from what is perceived in that paper as “low correlation between the textures in different bands” in multispectral images.
Another wavelet-transform-based NR methods is introduced in [2] based on the probability of the presence of features of interest [3], where denoising is carried out band-by-band taking into account the inter-band correlation. It is found that this method is performing well if the noise statistics are the same in all bands and is less suitable in the case where noise statistics varies from band to band. The inter-band correlation is also used in [4] to differentiate between the noise coefficients and the signal coefficients, which performs well in additive noise conditions.
Most of the hyperspectral/multispectral imagery NR methods perform well in fixed-variance additive noise environments. Unfortunately, real-life scenarios necessitate the existence of a signal-dependent noise component. In fact, at high SNR, the signal-dependent component becomes even more significant than the fixed-variance component because it is proportional to the signal amplitude.
In fact, the hyperspectral signal may vary dramatically from band to band with variable smoothness in different spectral regions, e.g. smoothness in the Visible and Near Infrared (VNIR) region compared to the smoothness in the Short-Wave Infrared (SWIR) region.
These and other considerations show that there is a need for better methods to increase SNR for such data and the signals derived from such data. Ideally, such methods would increase the SNR by reducing noise in the data or in the signal derived from such data.